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A classical reduced order model (ROM) for dynamical problems typicallyinvolves only the spatial reduction of a given problem. Recently, a novelspace-time ROM for linear dynamical problems has been developed, which furtherreduces the problem size by introducing a temporal reduction in addition to aspatial reduction without much loss in accuracy. The authors show an order of athousand speed-up with a relative error of less than 0.00001 for a large-scaleBoltzmann transport problem. In this work, we present for the first time thederivation of the space-time Petrov-Galerkin projection for linear dynamicalsystems and its corresponding block structures. Utilizing these blockstructures, we demonstrate the ease of construction of the space-time ROMmethod with two model problems: 2D diffusion and 2D convection diffusion, withand without a linear source term. For each problem, we demonstrate the entireprocess of generating the full order model (FOM) data, constructing thespace-time ROM, and predicting the reduced-order solutions, all in less than120 lines of Python code. We compare our Petrov-Galerkin method with thetraditional Galerkin method and show that the space-time ROMs can achieveO(100) speed-ups with O(0.001) to O(0.0001) relative errors for these problems.Finally, we present an error analysis for the space-time Petrov-Galerkinprojection and derive an error bound, which shows an improvement compared totraditional spatial Galerkin ROM methods.

Efficient space-time reduced order model for linear dynamical systems in Python using less than 120 lines of code